This is largely pointless, but may be interesting. Below is a process for calculating the function Z(x), for complex value x, where the real part = 0.5, and imaginary part > 0. The zeros of Z(x) are the zeros of the Riemann zeta function, for values with real part = 0.5.
https://en.wikipedia.org/wiki/Z_function to see an explanation of the formula: http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.html NB. Calculation of Z(x), estimate of zeta(x) for x = a +ib where b = 0.5 NB. Riemann-Siegel formula. (from Riemann Zeta Function, H.M Edwards) C0 =: 0.38268343236508977173, 0.43724046807752044936, 0.13237657548034352333, _0.01360502604767418865, _0.01356762197010358088, _0.00162372532314446528, 0.00029705353733379691,0.00007943300879521469, 0.00000046556124614504, _0.00000143272516309551, _0.00000010354847112314, 0.00000001235792708384, 0.00000000178810838577, _0.00000000003391414393, _0.00000000001632663392 C1 =: 0.02682510262837535, _0.01378477342635185, _0.03849125048223508, _0.00987106629906208, 0.00331075976085840, 0.00146478085779542, 0.00001320794062488,_0.00005922748701847, _0.00000598024258537, 0.00000096413224562, 0.00000018334733722 C2 =: 0.005188542830293, 0.000309465838807, _0.011335941078299, 0.002233045741958,0.005196637408862, 0.000343991440762, _0.000591064842747, _0.000102299725479, 0.000020888392217,0.000005927665493, _0.000000164238384, _0.000000151611998 C3 =: 0.0013397160907, _0.0037442151364, 0.0013303178920, 0.0022654660765,_0.0009548499998, _0.0006010038459, 0.0001012885828, 0.0000686573345, _0.0000005985366,_0.0000033316599, _0.0000002191929, 0.0000000789089, 0.0000000094147 C4 =: 0.00046483389, _0.00100566074, 0.00024044856, 0.00102830861,_0.00076578609, _0.00020365286, 0.00023212290, 0.00003260215, _0.00002557905, _0.00000410746,0.00000117812, 0.00000024456 pi2 =: +: o. 1 theta =: 3 : 0 assert. y > 0 p1 =. ( * -:@:^.@:(%&pi2)) y p2 =. --: y p3 =. -(o. 1) % 8 p4 =. 1 % (48 * y) p5 =. 7 % (5760 * y ^ 3) p1 + p2 + p3 + p4 + p5 ) summation =: 3 : 0 "1 v =. >0{y m =. >1{y thetav =. theta v ms =. >: i. m cos =. 2&o. s =. +:@:(^&_0.5) * cos@:(thetav&-@:(v&*@:^.)) +/ s ms ) remainderterms =: 3 : 0 im =. y v =. %: pi2 %~ y N =. <. v p =. v - N q =. 1 - 2 * p final =. 0 mv0 =. +/ (({&C0) * (q&^@:+:)) i. # C0 final =. mv0 mv1 =. +/ (({&C1) * (q&^@:+:@:>:)) i. # C1 mv1 =. mv1 * ((im % pi2) ^ _0.5) final =. final + mv1 mv2 =. +/ (({&C2) * (q&^@:+:)) i. # C2 mv2 =. mv2 * ((im % pi2) ^ _1) final =. final + mv2 mv3 =. +/ (({&C3) * (q&^@:+:@:>:)) i. # C3 final =. final + mv3 mv4 =. +/ (({&C4) * (q&^@:+:)) i. # C4 final =. final + mv4 final * (_1 ^ <: N) * ((im % pi2) ^ _0.25) ) z =: 3 : 0 "0 im =. y v =. %: pi2 %~ y N =. <. v p =. v - N v1 =. summation im ; N v2 =. remainderterms im v1 + v2 ) load 'plot' x =: steps 0.1 100 1000 plot x; z x On the graph you can see the zeros... and verify using the known zeros of the zeta function here http://www.dtc.umn.edu/~odlyZko/zeta_tables/zeros2 The first one is at about 14.13, and they appear in no particular pattern(?) from there. Anyway, as I said, this is largely pointless, but may interest some. Thanks, Jon ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
